Problem: Simplify the following expression: $y = \dfrac{8x^2- 31x- 45}{8x + 9}$
Explanation: First use factoring by grouping to factor the expression in the numerator. This expression is in the form ${A}x^2 + {B}x + {C}$ First, find two values, $a$ and $b$ , so: $ \begin{eqnarray} {ab} &=& {A}{C} \\ {a} + {b} &=& {B} \end{eqnarray} $ In this case: $ \begin{eqnarray} {ab} &=& {(8)}{(-45)} &=& -360 \\ {a} + {b} &=& &=& {-31} \end{eqnarray} $ In order to find ${a}$ and ${b}$ , list out the factors of $-360$ and add them together. Remember, since $-360$ is negative, one of the factors must be negative. The factors that add up to ${-31}$ will be your ${a}$ and ${b}$ When ${a}$ is ${9}$ and ${b}$ is ${-40}$ $ \begin{eqnarray} {ab} &=& ({9})({-40}) &=& -360 \\ {a} + {b} &=& {9} + {-40} &=& -31 \end{eqnarray} $ Next, rewrite the expression as $({A}x^2 + {a}x) + ({b}x + {C})$ $ ({8}x^2 +{9}x) + ({-40}x {-45}) $ Factor out the common factors: $ x(8x + 9) - 5(8x + 9)$ Now factor out $(8x + 9)$ $ (8x + 9)(x - 5)$ The original expression can therefore be written: $ \dfrac{(8x + 9)(x - 5)}{8x + 9}$ We are dividing by $8x + 9$ , so $8x + 9 \neq 0$ Therefore, $x \neq -\frac{9}{8}$ This leaves us with $x - 5; x \neq -\frac{9}{8}$.